Asymptotic Behavior of Self-Normalized Trimmed Sums: Nonnormal Limits

Hahn, Marjorie G.; Weiner, Daniel C.

doi:10.1214/aop/1176989937

Cited by 6 publications

(10 citation statements)

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“…Of the three cases, (3-9) is of most interest to us since it pertains to asymptotic normality. However, we should remark that (3)(4)(5)(6)(7)(8)(9)(10)(11) is an analogue of the result of Darling [3] on the dominance of the maximal summand when the tails of the underlying distribution are slowly varying.…”

Section: P( Max Y T £))mentioning

confidence: 98%

“…This is the case studied for example in Csorgo, Haeusler, and Mason [2], Griffin and Pruitt [6], Hahn, Kuelbs and Weiner [7] and Hahn and Weiner [8]. Further references and a discussion of other types of trimming can be found in any of the above references.…”

Section: P H I L I P S G R I F F I N and D A V I D M M A S O Nmentioning

confidence: 99%

“…It is an interesting problem to determine how the behaviour of nH(u n (a.)) effects that of T /^(r n) <r " >^n2 m ^e case (3)(4)(5)(6)(7)(8)(9)(10). This challenge has been taken up by Hahn and Weiner [8] who relate convergence in measure of nH(u n (a)) to a finite non-zero constant, to convergence of (<S n (r n )| (r » ) Z-1 |, F n (r n )| (r » ) Z; 1 |).…”

Section: P( Max Y T £))mentioning

confidence: 99%

See 2 more Smart Citations

On the asymptotic normality of self-normalized sums

Griffin

Mason

1991

Math. Proc. Camb. Phil. Soc.

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Let X1, …, Xn be a sequence of non-degenerate, symmetric, independent identically distributed random variables, and let Sn(rn) denote their sum when the rn largest in modulus have been removed. We obtain necessary and sufficient conditions for asymptotic normality of the studentized version of Sn(rn), and compare this to the condition for asymptotic normality of the scalar normalized version. In particular, when rn = r these conditions are the same, but when rn → ∞the former holds more generally.

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Section: P( Max Y T £))mentioning

confidence: 98%

Section: P H I L I P S G R I F F I N and D A V I D M M A S O Nmentioning

confidence: 99%

Section: P( Max Y T £))mentioning

confidence: 99%

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On the asymptotic normality of self-normalized sums

Griffin

Mason

1991

Math. Proc. Camb. Phil. Soc.

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“…The vast majority of this literature concerns independent data (e.g. Stigler 1973, Csörg½ o et al 1986, Hahn and Weiner 1992 and in a rare case mixing data with …nite variance (Hahn et al 1987).…”

Section: Heavy Tailed Nonlinear Distributed Lagsmentioning

confidence: 99%

“…1, no matter how heavy tailed X t is (Section 3). The array fX n;t g not only serves to approximate location for heavy tailed data, it forms the asymptotic foundation for a class of robust estimators, including self-normalized tail-trimmed sums (Pruitt 1985, Hahn et al 1990, Hahn and Weiner 1992, Hill 2009b and Generalized Method of Tail-Trimmed Moments (Hill and Renault 2010). The primary contribution of this paper is a set of new dependence notions that permit a broad Gaussian central limit theory for tail and tail-trimmed arrays of linear and nonlinear distributed lags and random volatility processes.…”

Section: 3mentioning

confidence: 99%

Tail and Nontail Memory With Applications to Extreme Value and Robust Statistics

Hill¹

2011

Econom. Theory

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New notions of tail and non-tail dependence are used to characterize separately extremal and non-extremal information, including tail log-exceedances and events, and tail-trimmed levels. We prove Near Epoch Dependence (McLeish 1975, Gallant andWhite 1988) and L 0 -Approximability (Pötscher and Prucha 1991) are equivalent for tail events and tail-trimmed levels, ensuring a Gaussian central limit theory for important extreme value and robust statistics under general conditions. We apply the theory to characterize the extremal and nonextremal memory properties of possibly very heavy tailed GARCH processes and distributed lags. This in turn is used to verify Gaussian limits for tail index, tail dependence and tail trimmed sums of these data, allowing for Gaussian asymptotics for a new Tail-Trimmed Least Squares estimator for heavy tailed processes.

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